Calculus I · Unit 3A · practice

Unit 3A Cumulative Practice

Practice method

Work in three passes

First, classify. Name the integral output and structural method before writing algebra. This separates a recognition error from a calculation error.

Second, solve without the key. Record a complete attempt, including bounds, constants, domains, units, or interpretation when the prompt asks for them.

Third, reveal one answer at a time. Compare the first line where your work differs, close the answer, and redo that item from a blank start.

Cumulative Practice

Exercise

Find the antiderivative family for $5x^4-3x^{-2}+2e^x$ .

Answer reveal

Exercise 1 answer

Your setup or answer

Write a real attempt before opening the supplied answer.

Exercise

Recover $F$ if $F'(x)=\sin x+x^2$ and $F(0)=5$ .

Answer reveal

Exercise 2 answer

Your setup or answer

Write a real attempt before opening the supplied answer.

Exercise

Construct $L_4,R_4,M_4$ for $f(x)=1/x$ on $[1,3]$ .

Answer reveal

Exercise 3 answer

Your setup or answer

Write a real attempt before opening the supplied answer.

Exercise

Evaluate $\int_{-2}^2(x^3+4)dx$ by symmetry and FTC.

Answer reveal

Exercise 4 answer

Your setup or answer

Write a real attempt before opening the supplied answer.

Exercise

Find the average value of $x^2$ on $[0,3]$ .

Answer reveal

Exercise 5 answer

Your setup or answer

Write a real attempt before opening the supplied answer.

Exercise

Differentiate $\int_{\cos x}^{x^2}\sqrt{1+t^4}dt$ .

Answer reveal

Exercise 6 answer

Your setup or answer

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Exercise

Evaluate $\int 4x e^{2x^2}dx$ .

Answer reveal

Exercise 7 answer

Your setup or answer

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Exercise

Evaluate $\int x^2\ln x\,dx$ .

Answer reveal

Exercise 8 answer

Your setup or answer

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Exercise

Evaluate $\int\sin^3x\cos^4x\,dx$ .

Answer reveal

Exercise 9 answer

Your setup or answer

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Exercise

Evaluate $\int dx/(x^2-1)$ on an interval avoiding the singularities.

Answer reveal

Exercise 10 answer

Your setup or answer

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Exercise

Approximate $\int_0^2\sqrt{1+x^3}dx$ by the trapezoidal rule with $n=4$ .

Answer reveal

Exercise 11 answer

Your setup or answer

Write a real attempt before opening the supplied answer.

Exercise

Determine whether $\int_0^1x^{-4/5}dx$ converges.

Answer reveal

Exercise 12 answer

Your setup or answer

Write a real attempt before opening the supplied answer.

After the explanation

Use the section idea

Reading lens

Mixed integral work tests recognition: classify the output and structure before committing to a method.

Mental model

A complete response carries setup, method, computation, bounds or constants, units, interpretation, and an independent verification.

Decision

Attempt the entire problem first, then use one answer at a time to locate the earliest reasoning decision that needs repair.

Common trap

Reading a key before modeling the problem or treating every mismatch as algebra when the first error was conceptual.

Check yourself

Can you reproduce the reasoning without the key and explain why the final form fits the question?

Source & rights

Original instruction with traceable references.

BetterGrades-original; no direct adaptation declared in the verified handoff.

Reference textbooks remain rights-separated and are not published as application assets. Any direct adaptation requires separate identification and attribution.